Section 1
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Cards (75)
Section 1
(50 cards)
1
f'(x)+g'(x)
-cot(x)+C
If f and g are inverses of each other, g'(x)
f is continuous at x=c if...
Alternative Definition of a Derivative
f '(x) is the limit of the following difference quotient as x approaches c
sec(x)tan(x)
Global Definition of a Derivative
tan(x)+C
1
Extreme Value Theorem
If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b]. The global extrema occur at critical points in the interval or at endpoints of the interval.
ln(x)+C
uvw'+uv'w+u'vw
Rolle's Theorem
Let f be continuous on [a,b] and differentiable on (a,b) and if f(a)=f(b) then there is at least one number c on (a,b) such that f'(c)=0 (If the slope of the secant is 0, the derivative must = 0 somewhere in the interval).
-csc(x)+C
dy/dx
cf'(x)
f'(g(x))g'(x)
-sin(x)
L'Hopital's Rule
Combo Test for local extrema
If f'(c) = 0 and f"(c)<0, there is a local max on f at x=c. If f'(c) = 0 and f"(c)>0, there is a local min on f at x=c.
First Derivative Test for local extrema
f'(x)-g'(x)
ln(cscx+cotx)+C = -ln(cscx-cotx)+C
-ln(cosx)+C = ln(secx)+C
hint: tanu = sinu/cosu
Formula for Washer Method
Axis of rotation is not a boundary of the region.
Fundamental Theorem of Calculus #2
-cos(x)+C
Mean Value Theorem for integrals or the average value of a functions
sin(x)+C
Horizontal Asymptote
0
Point of inflection at x=k
sec(x)+C
x+c
Exponential growth (use N= )
Intermediate Value Theorem
If f is continuous on [a,b] and k is a number between f(a) and f(b), then there exists at least one number c such that f(c)=k
Fundamental Theorem of Calculus #1
The definite integral of a rate of change is the total change in the original function.
nx^(n-1)
Formula for Disk Method
Axis of rotation is a boundary of the region.
ln(secx+tanx)+C = -ln(secx-tanx)+C
Squeeze Theorem
cos(x)
sec²(x)
The position function OR s(t)
Area under a curve
Mean Value Theorem
The instantaneous rate of change will equal the mean rate of change somewhere in the interval. Or, the tangent line will be parallel to the secant line.
Critical Number
If f'(c)=0 or does not exist, and c is in the domain of f, then c is a critical number. (Derivative is 0 or undefined)
-csc²(x)
ln(sinx)+C = -ln(cscx)+C
Section 2
(25 cards)